3.291 \(\int \frac{\sqrt{b x+c x^2}}{(d+e x)^3} \, dx\)

Optimal. Leaf size=127 \[ \frac{\sqrt{b x+c x^2} (x (2 c d-b e)+b d)}{4 d (d+e x)^2 (c d-b e)}-\frac{b^2 \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{8 d^{3/2} (c d-b e)^{3/2}} \]

[Out]

((b*d + (2*c*d - b*e)*x)*Sqrt[b*x + c*x^2])/(4*d*(c*d - b*e)*(d + e*x)^2) - (b^2
*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])
/(8*d^(3/2)*(c*d - b*e)^(3/2))

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Rubi [A]  time = 0.242748, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{\sqrt{b x+c x^2} (x (2 c d-b e)+b d)}{4 d (d+e x)^2 (c d-b e)}-\frac{b^2 \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{8 d^{3/2} (c d-b e)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[b*x + c*x^2]/(d + e*x)^3,x]

[Out]

((b*d + (2*c*d - b*e)*x)*Sqrt[b*x + c*x^2])/(4*d*(c*d - b*e)*(d + e*x)^2) - (b^2
*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])
/(8*d^(3/2)*(c*d - b*e)^(3/2))

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Rubi in Sympy [A]  time = 24.9, size = 105, normalized size = 0.83 \[ \frac{b^{2} \operatorname{atan}{\left (\frac{- b d + x \left (b e - 2 c d\right )}{2 \sqrt{d} \sqrt{b e - c d} \sqrt{b x + c x^{2}}} \right )}}{8 d^{\frac{3}{2}} \left (b e - c d\right )^{\frac{3}{2}}} - \frac{\left (b d - x \left (b e - 2 c d\right )\right ) \sqrt{b x + c x^{2}}}{4 d \left (d + e x\right )^{2} \left (b e - c d\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2+b*x)**(1/2)/(e*x+d)**3,x)

[Out]

b**2*atan((-b*d + x*(b*e - 2*c*d))/(2*sqrt(d)*sqrt(b*e - c*d)*sqrt(b*x + c*x**2)
))/(8*d**(3/2)*(b*e - c*d)**(3/2)) - (b*d - x*(b*e - 2*c*d))*sqrt(b*x + c*x**2)/
(4*d*(d + e*x)**2*(b*e - c*d))

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Mathematica [A]  time = 0.375513, size = 121, normalized size = 0.95 \[ \frac{\sqrt{x (b+c x)} \left (\frac{b^2 \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{\sqrt{x} \sqrt{b+c x} (b e-c d)^{3/2}}+\frac{\sqrt{d} (b (d-e x)+2 c d x)}{(d+e x)^2 (c d-b e)}\right )}{4 d^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[b*x + c*x^2]/(d + e*x)^3,x]

[Out]

(Sqrt[x*(b + c*x)]*((Sqrt[d]*(2*c*d*x + b*(d - e*x)))/((c*d - b*e)*(d + e*x)^2)
+ (b^2*ArcTan[(Sqrt[-(c*d) + b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/((-(c*d) +
b*e)^(3/2)*Sqrt[x]*Sqrt[b + c*x])))/(4*d^(3/2))

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Maple [B]  time = 0.016, size = 1963, normalized size = 15.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)^(1/2)/(e*x+d)^3,x)

[Out]

1/2/e/d/(b*e-c*d)/(d/e+x)^2*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^
(3/2)+1/4*e/d^2/(b*e-c*d)^2/(d/e+x)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*
d)/e^2)^(3/2)*b-1/2/d/(b*e-c*d)^2/(d/e+x)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(
b*e-c*d)/e^2)^(3/2)*c-1/4*e/d^2/(b*e-c*d)^2*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d
*(b*e-c*d)/e^2)^(1/2)*b^2+3/4/d/(b*e-c*d)^2*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d
*(b*e-c*d)/e^2)^(1/2)*b*c-1/2/e/(b*e-c*d)^2*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d
*(b*e-c*d)/e^2)^(1/2)*c^2+1/4/d/(b*e-c*d)^2*ln((1/2*(b*e-2*c*d)/e+c*(d/e+x))/c^(
1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))*c^(1/2)*b^2-3/4/
e/(b*e-c*d)^2*ln((1/2*(b*e-2*c*d)/e+c*(d/e+x))/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/
e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))*c^(3/2)*b+1/2/e^2*d/(b*e-c*d)^2*ln((1/2*(b*e-2
*c*d)/e+c*(d/e+x))/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(
1/2))*c^(5/2)-1/8/d/(b*e-c*d)^2/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+
(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e
+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*b^3+5/8/e/(b*e-c*d)^2/(-d*(b*e-c*d)/e^2)^(1
/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(
d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*b^2*c-1/e^2*d/(b
*e-c*d)^2/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+
2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(
1/2))/(d/e+x))*b*c^2+1/2/e^3*d^2/(b*e-c*d)^2/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(
b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-
2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))*c^3-1/4*e/d^2/(b*e-c*d)^2*c*(c
*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x*b+1/2/d/(b*e-c*d)^2*c^
2*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)*x-1/2/e*c/d/(b*e-c*d
)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2)-1/4/e*c^(1/2)/d/(b*e
-c*d)*ln((1/2*(b*e-2*c*d)/e+c*(d/e+x))/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x
)-d*(b*e-c*d)/e^2)^(1/2))*b+1/2/e^2*c^(3/2)/(b*e-c*d)*ln((1/2*(b*e-2*c*d)/e+c*(d
/e+x))/c^(1/2)+(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))-1/2/e^
2*c/(b*e-c*d)/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e
+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^
2)^(1/2))/(d/e+x))*b+1/2/e^3*c^2*d/(b*e-c*d)/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(
b*e-c*d)/e^2+(b*e-2*c*d)/e*(d/e+x)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-
2*c*d)/e*(d/e+x)-d*(b*e-c*d)/e^2)^(1/2))/(d/e+x))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x)/(e*x + d)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.252006, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, \sqrt{c d^{2} - b d e} \sqrt{c x^{2} + b x}{\left (b d +{\left (2 \, c d - b e\right )} x\right )} -{\left (b^{2} e^{2} x^{2} + 2 \, b^{2} d e x + b^{2} d^{2}\right )} \log \left (\frac{2 \,{\left (c d^{2} - b d e\right )} \sqrt{c x^{2} + b x} + \sqrt{c d^{2} - b d e}{\left (b d +{\left (2 \, c d - b e\right )} x\right )}}{e x + d}\right )}{8 \,{\left (c d^{4} - b d^{3} e +{\left (c d^{2} e^{2} - b d e^{3}\right )} x^{2} + 2 \,{\left (c d^{3} e - b d^{2} e^{2}\right )} x\right )} \sqrt{c d^{2} - b d e}}, \frac{\sqrt{-c d^{2} + b d e} \sqrt{c x^{2} + b x}{\left (b d +{\left (2 \, c d - b e\right )} x\right )} +{\left (b^{2} e^{2} x^{2} + 2 \, b^{2} d e x + b^{2} d^{2}\right )} \arctan \left (-\frac{\sqrt{-c d^{2} + b d e} \sqrt{c x^{2} + b x}}{{\left (c d - b e\right )} x}\right )}{4 \,{\left (c d^{4} - b d^{3} e +{\left (c d^{2} e^{2} - b d e^{3}\right )} x^{2} + 2 \,{\left (c d^{3} e - b d^{2} e^{2}\right )} x\right )} \sqrt{-c d^{2} + b d e}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x)/(e*x + d)^3,x, algorithm="fricas")

[Out]

[1/8*(2*sqrt(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x)*(b*d + (2*c*d - b*e)*x) - (b^2*e^2
*x^2 + 2*b^2*d*e*x + b^2*d^2)*log((2*(c*d^2 - b*d*e)*sqrt(c*x^2 + b*x) + sqrt(c*
d^2 - b*d*e)*(b*d + (2*c*d - b*e)*x))/(e*x + d)))/((c*d^4 - b*d^3*e + (c*d^2*e^2
- b*d*e^3)*x^2 + 2*(c*d^3*e - b*d^2*e^2)*x)*sqrt(c*d^2 - b*d*e)), 1/4*(sqrt(-c*
d^2 + b*d*e)*sqrt(c*x^2 + b*x)*(b*d + (2*c*d - b*e)*x) + (b^2*e^2*x^2 + 2*b^2*d*
e*x + b^2*d^2)*arctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/((c*d - b*e)*x)))/
((c*d^4 - b*d^3*e + (c*d^2*e^2 - b*d*e^3)*x^2 + 2*(c*d^3*e - b*d^2*e^2)*x)*sqrt(
-c*d^2 + b*d*e))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x \left (b + c x\right )}}{\left (d + e x\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2+b*x)**(1/2)/(e*x+d)**3,x)

[Out]

Integral(sqrt(x*(b + c*x))/(d + e*x)**3, x)

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GIAC/XCAS [A]  time = 0.572232, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x)/(e*x + d)^3,x, algorithm="giac")

[Out]

sage0*x